GIFT  or 


JLl^t  BtttoprBtJg  ai  (tiiirjuju 


A    Comparison    of   Different    Line 

Geometric    Representations   for 

Functions   of   a   Complex 

Variable 


A  DISSERTATION 

SUBMITTED  TO  THE  FACULTY 

OF  THE 

OGDEN  GRADUATE  SCHOOL  OF  SCIENCE 

IN  CANDIDACY  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 

(department  of  mathematics) 


GLADYS  E.  C.  G^BBNS 


31fF  (Halltv^uU  Prraa 

GEORGE  BANTA  PUBLISHING  COMPANY 

MENASHA,  WISCONSIN 

1922 


(ZIt|r  IniDrrfiity  of  CHIitraQo 


A   Comparison   of    Different    Line 

Geometric    Representations   for 

Functions   of   a   Complex 

Variable 


A  DISSERTATION 

SUBMITTED  TO  THE  FACULTY 

OF  THE 

OGDEN  GRADUATE  SCHOOL  OF  SCIENCE 

IN  CANDIDACY  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 

(department  of  kathebcatics) 


BY 

GLADYS  E.  C.  GIBBENS 


OBORGE  BANTA  PUBLISHING  COMPANY 

MENASHA,  WISCONSIN 

1922 


\ 


T^^^ 


f>«^ 


/ 


TABLE  OF  CONTENTS 

Introduction 1 

I.     Extension  of  the  Method  of  Parallel  Planes 3 

II.     The  Method  of  Non-Parallel  Planes 6 

III.     A  Generalization  of  the  Method  of  the  Riemann  Sphere. .  .  12 


492863 


A   COMPARISON   OF   DIFFERENT  LINE-GEOMETRIC    REP- 
RESENTATIONS FOR  FUNCTIONS  OF  A  COMPLEX 
VARIABLE 

INTRODUCTION 

Wilczynski*  has  recently  given  two  methods  for  constructing  a 
congruence  of  lines  determined  by  a  functional  relation  between 
two  complex  variables,  which  enable  one  to  visualize  the  properties 
of  the  function  by  studying  the  properties  of  the  resulting  congruence. 
In  the  first  method,  the  two  complex  variables  are  represented  upon 
two  distinct  planes,  parallel  to  each  other  and  a  unit  apart,  the 
corresponding  coordinate  axes  for  the  two  planes  being  chosen 
parallel  to  each  other  in  such  a  way  that  the  two  origins  lie  upon  a 
line  perpendicular  to  the  two  planes.  If  the  points  of  the  first 
plane  are  joined  to  the  points  of  the  second  plane  which  correspond  to 
them  by  means  of  a  given  functional  relation  w=F(z),  a  two-parame- 
ter family,  or  a  congruence,  of  straight  lines  is  obtained.  These 
congruences  have  certain  characteristic  properties  which  hold  for  the 
totality  of  analytic  functions,  and  in  addition,  of  course,  special 
properties  which  depend  upon  the  choice  of  the  particular  function 
F{z).  The  developables  and  focal  sheets  of  such  congruences  are 
always  imaginary,  except  in  a  trivial  special  case,  but  some  interest- 
ing real  surfaces  are  closely  associated  with  them. 

The  second  method  of  representation  makes  use  of  a  Riemann 
sphere.  The  two  complex  variables  are  projected  upon  the  same 
sphere,  and  points  of  the  sphere  corresponding  to  each  other  by  means 
of  the  function  w  =  F{z)  are  joinajj  by  lines.  The  congruences 
obtained  in  this  way  always  have  real  focal  sheets  and  developables, 
and  are  therefore  more  interesting  than  those  obtained  by  the  first 
method. 

It  can  be  seen  at  once  that  there  are  other  methods  of  constructing 
congruences  of  lines  in  connection  with  a  relation  w  =  F(2),  these  other 

*  E.  J.  WiLCZYNSKi:  "Line-geometric  representations  for  functions  of  a  complex 
variable,"  Transactions  of  the  American  Mathematical  Society,  Vol.  XX  (1919), 
pp.  271-298. 


2  Line-Geometric  Representations 

methods  imposing  less  drastic  restrictions  upon  the  planes  or  spheres 
upon  which  the  variables  are  represented.  It  is  the  purpose  of  the 
present  paper  to  consider  properties  of  congruences  which  are  ob- 
tained from  such  generalizations  of  the  above  methods.  In  section 
I  we  shall  study  the  congruences  resulting  when  the  planes  of  refer- 
ence are  kept  parallel,  but  when  the  coordinate  axes  in  the  two 
planes  are  given  arbitrary  positions.  We  shall  find,  as  might  be 
expected,  that  these  congruences  are  not  essentially  distinct  from 
those  corresponding  to  the  special  case.  In  fact,  if  a  congruence  is 
constructed  by  considering  one  of  the  complex  variables  as  a  given 
function  of  the  other,  then  a  projectively  equivalent  congruence  may 
be  obtained  by  keeping  the  coordinate  axes  parallel,  and  considering 
in  place  of  the  given  function  one  closely  related  to  it,  a  rotated 
function,  the  angle  of  rotation  being  the  angle  between  corresponding 
axes. 

In  section  II  we  shall  consider  the  general  properties  of  those 
congruences  obtained  by  the  general  conformal  correspondence 
between  two  planes,  the  relative  positions  of  the  two  planes  in  space 
being  left  arbitrary.  It  will  be  found  that  this  general  theory  is 
included  essentially  in  that  special  case  in  which  the  two  planes  are 
perpendicular  to  each  other,  and  the  axes  occupy  certain  special 
positions.  Such  a  method  of  representation  has  a  serious  disadvan- 
tage. For  an  arbitrary  function,  it  is  impossible  to  predict  whether 
the  developables  and  focal  sheets  of  the  congruence  are  real  or 
imaginary.  Therefore  from  the  point  of  view  of  the  general  theory 
of  functions,  such  a  method  is  far  less  useful  than  the  method  of 
parallel  planes,  though  it  may  be  of  value  in  special  instances. 

Section  III  deals  with  an  extension  of  the  method  of  the  Riemann 
sphere.  The  two  complex  variables  will  be  projected  upon  two 
distinct  but  concentric  spheres.  As  in  section  II,  the  simpler  method 
furnishes  the  more  valuable  results.  The  method  of  concentric 
spheres  does  not  permit  us  to  make  a  general  statement  as  to  the 
reality  of  the  developables  and  focal  sheets  for  all  possible  functions 
w  =  F{z),  as  in  the  case  when  we  use  a  single  sphere.  In  the  more 
general  case,  the  properties  of  the  individual  function  play  an  essen- 
tial r61e  in  answering  such  questions. 

The  author  wishes  to  express  her  gratitude  to  Professor  Wilczynski 
for  his  constant  interest  and  helpful  suggestions  to  her  in  the  writing 
of  this  thesis. 


I.  EXTENSION  OF  THE  METHOD  OF  PARALLEL  PLANES 


Let  us  denote  by 


(z  =  x-\-iy,         Zo  =  x—iy 
\w  =  u-\-iv, 


(1)  \w  =  u-\-iv,        Wo  =  u—iv 

the  two  complex  variables  and  their  conjugates,  and  assume  the 
functional  relation 

w  =  F{z) 
which  implies  that 

Wq  =  Fo{Zq), 

where  Ff,  is  the  function  conjugate  to  F. 

Let  us  represent  the  point  P^  upon  the  ^T;-plane  of  a  system  of 
{77  5-axes  in  ordinary  cartesian  space,  letting  the  x-  and  y-a.xes  coincide 
with  the  $-  and  77-axes  respectively.  Then  the  space  coordinates  of 
P,are 

(2)  ^i  =  x,         r,i  =  y,  51  =  0. 

Now  let  us  represent  the  variable  w  upon  a  plane  parallel  to  the 
^-plane,  a  unit  above  it,  but  allow  the  real  and  imaginary  axes  in 
this  plane  to  be  in  an  arbitrary  position.  If  the  angle  between  the 
^-  and  w-axes  is  6,  and  if  the  coordinates  of  the  origin  of  the  complex 
numbers  u-{-iv  are  (a,  b,  1),  then  the  point  P„  will  have  the  coordi- 
nates 

(3)  ^2  =  M  cos  6—v  sin  d-{-a,  r]2  =  u  sin  d-\-v  cos  d-{-b,  j;2=1- 

Let  R  he  a.  region  of  the  2-plane  in  which  the  function  w  =  F{z)  is 
analytic,  and  let  us  join  each  point  of  this  region  to  the  corresponding 
points  P^.  If  the  function  is  n- valued,  where  n  is  finite,  there  will  be 
n  lines  of  the  congruence  through  each  point  of  R.  The  projective 
properties  of  the  congruence  defined  in  this  way  will  be  studied  by 
means  of  a  system  of  diflferential  equations  of  the  type'^ 

*  E.  J.  WiLCZYNSKi:  "One  parameter  families  and  nets  of  ruled  surfaces  and  a  new 
theory  of  congruences,"  Transactions  of  the  American  Mathematical  Society,  Vol. 
XXI  (1920),  pp.  157-206.  This  paper  will  hereafter  be  referred  to  as  "Ruled  surfaces 
and  congruences." 


Line-Geometric  Representations 

d^  .         d\  .         dfi  .         -   ,  - 

— -4-pii [-pi2  —  1  qu  X+qi2M  =  0 

dz^  dz  az 

dV  I         d\  d/ji 

— -+P21  — I-P22 hqii  X+q22/i  =  0 

dz^  dz  az 

W      lax  d\,         dfjL,,      ... 

—  =aii  — +ai2  — hbii  X+bia/x 
dzo  dz  dz 

^f^        o       ^^_l_o       ^'".LK       \_LK 
=  3.21 |-a22 t-b2l  X+b22M 

dzo  dz  dz 

Clearly  the  relation  which  we  have  assumed  between  the  space 
coordinate  system  and  the  real  and  imaginary  axes  of  the  z-plane 
involves  no  loss  of  generality.  We  may  also  assume  that  a  =  6  =  0, 
so  that  the  origin  of  coordinates  for  the  variable  w  is  on  the  g-axis, 
without  changing  the  projective  properties  of  the  congruence.  For 
make  the  projective  transformation  of  space 

(5)  l  =  ^-az,V  =  V-b  z,Z  =  Z 
then  the  coordinates  of  P,  and  P^, 

(6)  Ui=x  ;vi=y  ;?i=o 

1^2  =  **  cos  d—v  sin  d-{-a;  r\i  =  u  sin  d-\-v  cos  B-\-h\  52=  1 
will  become 

(7)  /ii  =  «  ,n\=y  |i  =  0 


,{2  =  M  cos  d—v  sin  d,r]2  =  u  sin  d-\-v  cos  6,  52=  1 
which  makes  a  =  6  =  0. 

By  means  of  (1),  we  can  introduce  into  (7)  the  variables  z,  Zo,  w, 
Wq.  We  find  the  following  homogeneous  cartesian  coordinates  for 
P.  and  P„: 


(8) 


Xi=-(z+zo),  ni-^^osd(w-{-Wo)-—sm  B{w-Wq), 

Xl=;r^(z-Zo),  /*2=;:Sin  6  (w;+Wo)+;:  COS  diw-Wa), 
It  I  It 

X8  =  0,  /xj=l 

.  X4=l,  /i4=l. 


Line-Geometric  Representations 


If  we  write 


(9) 


fa  =cos  d-^i  sin  d 
\ao  =  cos  d—i  sin  6, 


(8)  can  be  written  in  the  form 


(10) 


X3  =  0, 

1X4=1, 


p.- 

M4=l. 


If  in  particular,  0  =  0°,  so  that  the  u-  and  ac-  axes  are  parallel,  (10) 
reduces  to  the  special  case  considered  by  Wilczynski, 


(11) 


P,: 

Xi  =  -(z+2o), 

"Xl=;r.(z-2o), 
Zt 

X3  =  0, 

1X4=1, 


M2=2^(w;-Wo), 

At3=l, 


A  comparison  of  (10)  and  (11)  will  show  that  the  two  situations 
are  equivalent.  For  in  studying  the  totality  of  analytic  functions 
w  =  F(z),  in  (11),  among  them  will  be  included  those  derived  from 
a  particular  one  by  multiplying  it  by  the  rotating  factor  a  =  e^, 
giving  the  function  which  appears  in  (10).  Thus  the  projective 
properties  of  the  class  of  congruences  which  is  defined  by  the  totality  of 
all  analytic  functions  w  =  F{z)  by  the  method  of  parallel  planes,  are 
independent  of  the  relative  position  of  the  origins,  of  the  angle  between 
the  real  axes  of  the  two  complex  variables,  and,  of  course,  of  the  distance 
between  the  two  planes.  The  congruence  which  corresponds  to  an  indi- 
vidual function  F{z)  in  any  particular  representation  of  this  sort  cor- 
responds not  to  the  same  function  but  to  the  function  e^w  =  F{z),  if  the 
angle  between  the  recti  axes  of  the  two  planes  be  changed  by  6. 


II.     THE  METHOD  OF  NON-PARALLEL  PLANES 


Let  us  consider  now  the  case  in  which  the  planes  upon  which 
the  two  complex  variables  are  represented  are  not  parallel.  Then 
the  line  of  intersection  of  the  two  planes  will  be  a  proper  line,  which 
we  may  choose  as  the  ^-axis.  We  may  identify  the  ^77-  and  z-planes, 
and  choose  as  the  ?7-axis  a  line  which  passes  through  the  origin  of  the 
x^'-system.     Let  us  use  the  following  notations: 

ip  =the  angle  between  the  two  planes. 

01=  the  angle  between  the  ^-  and  x-axes. 

02  =  the  angle  between  the  %-  and  «-axes. 

(0, 61)  ^  the  coordinates  of  the  origin  0\  of  the  acy-system, 

(12)  \  with  respect  to  the  ^77-axes. 
(aj,62)  =  the  coordinates  of  the  origin  0%  of  the  «»-system, 

with  respect  to  a  system  composed  of  the  ^-axis 
and  the  line  of  intersection  of  the  7j:5-plane  with 
the  w-plane. 
Then  the  cartesian  coordinates  of  P,  and  P«,  are 

P.:  Pv>: 

{i  =  a;cos0i— y  sin  0],  ^2  =  wcos02— »sin  ^2+02, 

(13)  '  »7i  =  aj  sin  Qv\-y  cos  Bv\-h\,      772  =  cos  ^[m  sin  02+tJ  cos  B%-\-h^, 

j;i  =  0  z;2  =  sin  v?!"  sin  02+ f  cos  02+^2]." 

We  change  to  a  projectively  equivalent,  but  simpler  form  by 
means  of  the  transformation 

z     V  cos  ^  _        1 

^=f,        *)=»? — \ 5;  ?=-^ 1 

sm  ip  sm  ip 

which  is  admissible  since  the  z-  and  iv-  planes  are  assumed  to  be  non- 
parallel.  The  new  coordinates,  expressed  in  homogeneous  cartesian 
form,  are 


(14) 


P.- 

\x  —  x  cos  B\—y  sin  0i, 
\i  =  x  sin  Q\-\-y  cos  0i+6i, 

1X4=1, 


H\  =  u  cos  Bt—v  sin  02+^2, 

A»2  =  0, 

/U8=M  sin  B%-\-v  cos  ^2+62, 

/i4=l. 


Line-Geometric  Representations  7 

But  the  values  of  X*,  nk,  as  given  by  (13),  would  reduce  to  these 
same  values  for  ^'  =  90'*.  We  have  shown  therefore,  that  if  the  planes 
of  reference  are  non-parallel,  a  congruence  of  this  sort  is  projectively 
equivalent  to  one  obtained  from  it  by  rotating  the  w-plane  around  the 
line  of  intersection  of  the  two  planes  until  the  z-  and  w-planes  are 
perpendicular  to  each  other. 

If  we  use  the  notation 

(15)  j  a  =  cos  6i-\-i  sin  6i,  /S  =cos  62-\-i  sin  02, 

\ao  =  cos  6i  —  i  sin  di,  /So  =  cos  02— i  sin  02, 

and  introduce  into  (14)  the  complex  variables  given  by  (1),  we  have 
the  following  coordinates  for  Pg  and  /*«,: 

P  •  P  • 


(16) 


Xi=-(a2+aoSo),  /ii= -(/3w+/9oWo+2fl2), 

X2  =  y.(a2— aoZo+2*Oi),  . 

j^  ^Q*  fi3  =  Y-^Pw-fioiVo+2ib2), 


X4=l,  M4=l- 

By  an  argument  similar  to  that  used  in  section  I,  we  see  that  it  is 
not  necessary  to  consider  this  general  situation.  Equations  (16) 
should  also  have  been  obtained  if,  in  the  two  perpendicular  planes 
of  reference,  the  x-  and  u-  axes  had  been  taken  parallel  to  the  line  of 
intersection  of  the  planes,  while  the  variables  from  which  the  congru- 
ence was  constructed  were 

Z=az, 

W  =  ^w. 

This  amounts  to  a  transformation  of  both  independent  and  dependent 
variables,  rotating  them  through  angles  corresponding  to  the  angles 
between  the  individual  real  axes,  and  the  line  of  intersection  of  their 
planes.  Since  our  point  of  view  is  the  study  of  the  totality  of  all 
such  functional  relations,  the  more  special  case  will  suffice.  We 
may  assume  therefore,  without  loss  of  generality, 

a=ao  =  i8  =  /3o=l 


8 


Line-Geometric  Representations 


and  (16)  may  be  written 


(17) 


Xl  =  -(z  +  2o), 
X2=2".(2-2o+2t*i) 

X,  =  0, 


1X4=1, 


Ail  =  2(^+^0+202), 

/X2  =  0, 


M8=r-.(M'-Xfo+2>6j), 


M4=l. 


If  we  follow  the  line  Zo=  const,  in  the  z-plane,  we  obtain  a  ruled 
surface  of  the  congruence.  If  this  ruled  surface  is  a  developable,' 
the  pairs  (X,-,/i,-,«  =  1 ,  .  .  .4)  of  (17)  must  satisfy  the  relation 

^■,      ^,      X,     Mil  =  0     (»=1,2,3,4). 
dz         dz 


(18) 


Similarly,  if  the  family  of  ruled  surfaces  z  =  const,  consists  of  devel- 
opables  the  relation 


(180 


OZo  OZo 


M.|  =  0     («=1,  2,  3,  4) 


must  hold.     These  reduce  to  a  single  condition 

Wo'(z  —  w  —  a2-i-i(bi  —  b2))  =  0. 
If  Wo'=0,  or  2^^  =  const.,  the  congruence  reduces  to  a  bundle  of  lines 
through  a  point  on  the  w-plane.     If  the  second  factor  vanishes,  then 

iv-{-a2-\-ib2  =  z-\-ibi, 
which  is  a  special  linear  function,  and  represents  a  bundle  of  parallel 
lines,  perpendicular  to  the  line  of  intersection  of  the  two  planes,  and 
cutting  them  at  equal  distances  from  this  line.  For  all  other  func- 
tions, the  two  families  of  ruled  surfaces  z  =  const.,  Zo  =  const,  are  not 
developables. 

Let  us  now  derive  the  system  of  differential  equations  (4)  which 
the  coordinates  (17)  are  to  satisfy.  Since  the  coordinates  are  linear 
in  the  variables,  the  second  order  equations  can  be  found  at  once. 
They  are 

(19) 


dz^ 


^  =  0. 


dw' 


If  we  introduce  z  as  independent  variable  in  the  latter  equation  it 
becomes 


'  "Ruled  surfaces  and  congruences,"  p.  158. 


(20) 


Line-Geometric  Representations  9 

d2*     w'      dz 
The  coefficients  of  the  first  order  equations  may  be  found  by  the 
method    of    undetermined    coefficients.     The    complete    system    is 

^-!^=o,     ^-^  ^-^=0, 

3/i*  dz^     w'      dz 

d\  _Wo  —  z  -\-ai  —  i(bi-\-bi)  d\     I      w  —  Wo-\-2ib2  ^m  , 

dzo    ZQ—Wo—a2—i{bi  —  bi)dz     iv'    ZQ—wo  —  a2—i(bi  —  bi)   dz 
X-/X 
Zo—WQ—at  —  iibi  —  bi) 
1    dn_Z9—z  —  2ibi  d\     I    w  —  zo+ai-\-i(bi+bi)    , 

Wo' dzo    Zo  —  wo—at—i^bi  —  bt)    dz     w'  zo—Wo—a2—i(bi  —  bi) 
-mX 
Zo—WQ—a2—i{bi  —  bi) 

where  w'  and  Wo'  indicate  the  derivatives  of  these  functions  with 
respect  to  z  and  zq  respectively. 

The  first  step  in  the  reduction  of  system  (20)  to  the  canonical  form 
shows  the  disadvantage  of  this  method  of  constructing  a  congruence. 
Instead  of  the  two  given  planes  of  reference,  the  two  focal  sheets  of 
the  congruence  could  be  introduced  as  new  surfaces  of  reference. 
This  involves  a  change  of  dependent  variables,  the  new  ones  being 
obtained  from  the  linear  factors  of  the  quadratic  covariant* 

(21)  asi  X^— (flu— (122)  X/i— ai2M^ 

In  the  parallel  plane  representation,  these  two  points  on  the  lines 
of  the  congruence  were  always  imaginary,  for  all  non-trivial  functional 
relations,  and  therefore  the  focal  loci  were  always  imaginary  surfaces. 
A  single  example  will  suffice  to  show  that  in  the  present  representa- 
tion, for  this  function,  the  focal  points  are  real  on  some  lines  of  the 
congruence,  and  imaginary  on  others.  Hence  no  general  statement 
can  be  made  about  the  reality  of  the  focal  surfaces  for  an  arbitrary 
functional  relation.     In  our  case  the  co variant  (21)  has  the  value 

C22)  /"''^o'(^~^o+2«6i)X'-f  {w'[wo-2+a2-*(*i+*2)] 
\—Wo'[w—Zo-{-a2-\-i(bi-\-b2)]}\n-\-(w  —  WQ-}-2ib2)fi^. 

This  is  a  quadratic  form  with  imaginary  coefficients.     Then  the 
*  "Ruled  surfaces  and  congruences,"  p.  183. 


10  Line-Geometric  Representations 

character  of  its  factors  will  be  determined  by  the  sign  of  its  discrimi- 
nant 

.23xl'A=  lw'ko-z+a2-«(6i+^2)]-W'o'[w-Zo+as+»(*i+Ml' 
U    -4w'wo'[z-Zo-h2ibi][w—wo+2ibi]. 

Now  let  us  consider  the  function 

w  =  e' 
and  set  02=61  =  62  =  0.     Then  A  becomes 

A  =  [zoe"-zeY-^'-^"iz-Zo)(e'-e''): 
Let  2  vary  over  pure  imaginary  values,  z  =  iy.     Then  we  have 

A=  —  4y[y  cos^  y  —  4  sin  y] 
which  has  the  special  values 

A  =  87r  for  3'  =  - 
A  =  —  47r2  for  y  =  T. 

Since  the  sign  of  A  changes  from  positive  to  negative,  the  factors  of 
(21)  are  sometimes  real  and  sometimes  imaginary.  Therefore  some 
lines  of  the  congruence  have  real  focal  points  while  others  have 
imaginary  focal  points. 

A  further  disadvantage  of  this  method  appears  when  we  try  to 
introduce  as  new  independent  parameters  those  which  correspond 
to  the  developables  of  the  congruence.  This  involves  the  integration 
of  the  partial  differential  equation* 

(24)    I  —  I  -(011+^22) +  {ana22  — ai2a2)  I  —  I  =0 

\dzo/  dz  dzo  \dz/ 

which  in  the  present  case  assumes  the  form 

w'[zQ—wo—a2—i{bi  —  b2)]\  —  )  —  {w'[wQ—z+a2—i(bi-{-b3)] 

\dzo/ 


(25) 


-\-Wo'[w—Z(,-\-ai-^i(bi-\-b2)]  ] \-Wo'[z-w—a2-\-i(bi  —  bi)] 

dz  dzo 


\dzo/ 


0 


which  does  not  seem  to  admit  of  any  simple  method  of  integration. 

From  the  point  of  view  of  obtaining  geometric  properties  common 

to  all  analytic  functions,  the  method  of  parallel  planes  seems  to  be 

*  ''Ruled  surfaces  and  congruences,"  p.  187. 


Line-Geometric  Representations  11 

more  powerful,  though  the  present  more  general  method  may  be 
advantageous  for  special  functions.  In  fact  when  02  =  61  =  ^2  =  0, 
for  such  simple  functions  as  u>  =  z'*  with  n  real;  w  =  cz  where  c  is  a 

complex  constant;  and  for  the  class  of  linear  functions'  w  = 

cz-\-d 

a,  b 


when  a,  b,  c,  d  are  real  with 


<0,  the  focal  points  are  always 


c,  d 

real,  while  in  the  parallel  plane  representation  they  are  always  im- 
aginary. 

'  Compare  A.  Emch:  "On  the  rectilinear  congruence  realizing  the  circular  trans- 
formation of  one  plane  into  another,"  Annals  of  Mathematics,  2nd  series.  Vol.  M  13, 
(1911-12),  pp.  155-160. 


III.     A  GENERALIZATION  OF  THE  METHOD  OF  THE 
RIEMANN  SPHERE 


Let  us  choose  the  ^77-plane  as  the  common  plane  of  the  two 
complex  variables.     Project  the  variables 

z  =  x-\-iy 

upon  a  sphere  Si  of  radius  ri  with  its  center  at  the  origin,  using  as 
center  of  projection  the  point  {0,  0,  ri).  Also  project  the  second 
variable 

w  =  u-\-iv 

upon  a  concentric  sphere  52  with  radius  r2  from  the  point  (0,  0,  rj). 
If  the  radii  fi  and  r^  are  equal,  the  spheres  will  coincide  and  the 
situation  will  reduce  to  that  considered  by  Wilczynski.  A  line  of  the 
congruence  is  obtained  by  joining  a  point  P,  of  Si  to  the  points  P» 
on  52  which  correspond  to  it  by  means  of 

w  =  Fiz). 

The  correspondence  between  the  two  spheres  is  of  course  conformal 
since  the  null  lines  on  the  two  spheres  correspond  to  each  other. 
While  this  is  not  the  most  general  extension  which  can  be  made  of 
Wilczynski's  method,  it  puts  into  evidence  very  clearly  the  diflScul- 
ties  which  arise  from  any  such  generalization. 

The  coordinates  of  P,  and  P„  can  be  found  at  once.     The  line 
joining  the  points  (x,  y,  0)  and  (0,  0,  fi)  will  cut  the  sphere 

Si  e+-^+Z^  =  ri^ 

in  a  point  whose  coordinates  are 

(26)     fi= ,    rii  = ,   gi  = . 

x^-\-y^-^ri^  x2+/+ri2'  ;r2+/+r*i 


Line-Geometric  Represmtations 


13 


Similarly,  the  coordinates  of  P.,  the  intersection  of  S%  with  the  line 
joining  («,  v,  o)  and  (o,  o,  rj)  are 


(27)   {,= 


Inht 


K'+r'+fj* 


i7j  = 


2rs*T; 


tt'+t^'+rj' 


?J  = 


The  homogeneous  cartesian  coordinates  of  the  points  may  therefore 
be  written  in  the  form 


(28) 


P.: 

\i  =  2ri% 
\i  =  2ri*y, 
X8  =  r,(a:2+y2-r,2), 


P.: 

fi2  =  2r2h, 


or  after  introducing  the  complex  variables  of  (1) 


(29) 


P.- 

Xi  =  ri2(2+2o), 

X2=-«V(Z  — Zo), 

"  X3  =  ri(22o— ri^), 

X4  =  2Zo+flS 


P^: 

ti\  =  r^{w-\-WQ), 
H2=—ir  2^(10  — Wo), 
ti3  =  r2(wwQ—r2^), 
tiA  =  wu>o-\-r2^. 


The  conditions  (18)  and  (18')  that  the  two  families  of  ruled  surfaces 
2  =  const.,  zo=  const,  shall  be  developables,  reduces  to 

w'{riw  —  r^)  =  0 

which  can  be  interpreted  immediately.     If  w'  =  0,   or  w  =  const., 

the  congruence  becomes  a  bundle  of  lines  through  a  point  of  52.     If 

riW—r2Z  =  0  or 

r22 
w=—, 

ri 

the  congruence  reduces  to  the  bundle  of  lines  through  the  common 
center  of  the  spheres. 

For  all  other  functions,  the  coordinates  (29)  will  satisfy  a  system 
of  differential  equations  of  the  form  (4).  In  fact,  the  second  order 
equations  are  identical  with  those  obtained  in  section  II.  The 
coefficients  of  the  first  order  equations  have  the  following  values: 


14  Line-Geometric  Representations 

(ri-r2)(rir24-zW 


(30) 


an  = 


(ri-\-r2){riWo—r^oy' 


ri' 

ais= — . 

w  r2^ 


j  {ri-\-ri) (riWo—rj^o) (riw  —  fzz)  +  (fi  —  fa) (r ifa+gM'o) (rir2-\-Zow) 
I  (ri+f2)(fiWo-r2Zo)' 

;,^^^  _  \r2(ri-\-r2)(riWQ—r2Zo)-\-wo(ri  —  r2)(rir2-{-zwo) 
1  (ri+r2)(riWo— fjzo)^  J 

J  ^^^i^\ri(ri-\-r2)(riWQ—r2Zo)-{-Zo{ri-r2)(rir2-{-zwo)] 
ri^]  (ri+r2)(ri«'o-^2Zo)^  J' 


021=  — 


w^r^ 


|(fi+r2)(yiWo— y22o)(riW-r22)  — (ri-r2)(rir2+2Wo)(rir2+2ow)  1 
I  (ri+r2)(ri«^o-f2Zo)^  J 

^'  [(ri+r2)(riWo-r22o)Y 

jj ^o^^2'[r2(fi+r2)(fi«>o— f22o)4-Wo(ri-r2)(rir2+Zow)] 

^1^    I  (fi+r2)(riWo-r2Zo)'*  J     , 

^22=  -n,^Ar\{r\^-r2){rxWQ-r2Z^^rZQ{rx-r^{rxr2^■ZQw)\ 
\  (ri+r2)(riWo-r2Zo)*  J 

The  covariant  (21)  which  determines  the  coordinates  of  the  focal 
points  of  the  lines  of  the  congruence,  becomes 

O' Wr2*  { (''i+^'z)  (''i«'o  —  r2Zo)  (riW' -  f 22)  -  (ri  -  r2)  (rir2+za'o) 
(rif2+Zow))X2 
(31)  \  +riW(ri-r2)  {w'(rir24-zwo)2+w;o'(rif2+Zoa/)2)XM 

-ri''{(ri+r2)(ri«'o-r22o)(ri«'— r2,)  +  (ri-r2)(»'if2+2Wo) 
(rir2+2o«;)))u^ 

whose  coefl&cients  are  real  and  its  discriminant  is  equal  to 

(32)—  =  (ri-r2)  V(rir2+2W'o)'- W(rir,+2ow)^P+4tt;W(ri+r2)2 
riV2* 

(riWo—  r22o)^(f !«'  —  ^22)^^. 

If  we  exclude  the  case  ri  =  r2,  for  which  the  preceding  results  all 
become  simple,  we  see  that  the  first  term  in  A  is  always  negative, 


Line-Geometric  Representations  15 

while  the  second  is  always  positive.     The  following  example  shows 
that  for  a  given  function,  A  can  be  changed  in  sign  as  z  varies.     Let 

•w  —  az, 

where  a  is  a  complex  constant.     Then  (32)  gives  the  value 

=  (r  1  —  r  j)'(a  —  aoYiaat^aho^y 

ri*rt* 

-\-2(aa^W)[2{ri-\-r2y(riao-r2yiria-r2y-rMHa-aoyiry-r2y] 

-^ri*r2*{rr-r2y{a-aoy 

which  is  a  quadratic  form 

A(aa^Wy-\-2B{aaozW)+C 

with  real  coefficients  A<0,  B>0,C<0  and  with 

B^-AC^iri+riYiriao-riYiria-r^y 
-  (r  i^  -  rz^y  {r  lOo— rzyiria  — Tzy  (a  — aoy 

which  is  always  positive.     Then  there  are  two  positive  real  values  of 
aa^^ho^  for  which  A  will  vanish.     If  we  indicate  them  by  pi  and  p2, 

Pi^{aa(fih^)  <p2 

represents  the  closed  region  for  which  the  lines  of  the  congruence 
have  real  focal  points. 

The  situation  here,  then,  is  similar  to  that  in  section  II.  The 
reality  of  the  focal  sheets  and  developables  of  the  congruence 
depends  upon  the  special  function  under  consideration.  In  the  par- 
ticular cases  studied  by  Wilczynski,  the  reality  conditions  are  inde- 
pendent of  the  particular  functional  relation.  From  the  point  of 
view  of  a  general  theory,  then,  there  seem  to  be  serious  disadvan- 
tages connected  with  any  attempt  to  generalize  these  two  methods 
for  representing  a  functional  relation  by  means  of  a  congruence. 


16  Line-Geometric  Representations 


VITA 

Gladys  Elizabeth  Corson  Gibbens  was  born  in  New  Orleans, 
Louisiana,  on  January  21,  1893.  Her  early  education  was  received 
in  the  public  schools  and  the  Ursuline  Academy  of  that  city.  She 
then  attended  Newcomb  College,  from  which  institution  she  re- 
ceived the  degree  of  B.A.  in  1914.  During  the  years  1914-1917  she 
held  a  teaching  fellowship  in  mathematics  at  Newcomb  College,  and 
at  the  same  time  studied  at  Tulane  University,  being  granted  the 
degree  of  M.A.  in  1916. 

She  studied  at  the  University  of  Chicago  for  nine  quarters  during 
the  years  1917-1920,  holding  a  fellowship  in  the  department  of 
mathematics  for  the  last  two  years  of  that  period.  She  has  had 
courses  under  Professors  Moore,  Dickson,  Bliss,  Wilczynski  and 
Moulton,  to  all  of  whom  she  wishes  to  express  her  appreciation  of 
their  helpful  interest.  Her  special  thanks  are  due  to  Professor 
Wilczynski,  whose  teaching  has  been  a  constant  inspiration  in  the 
writing  of  this  thesis. 


UNIVERSIIY  OF  CALIFORNU  LIBRARY 
^boolcisDUEon^elastdaeesta„.ped  below. 


LD21-ioom.l2.'46(A20l28l6)4120 


Photomount 

Pamphlet 

Binder 

Gaylord  Bros. 

Makers 

Syracuse,  N.  Y. 

PIT.  JU  :i,  1901 


492H(iC: 


-%^ie^^ 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


